At the heart of our approach lies a classification of certain Anosov actions of higher rank abelian groups, and building additional invariant structures which lead to classification. Connecting actions of semisimple groups and their lattices with those of higher rank abelian subgroups goes all the way back to Hurder's proof of local rigidity of the action of \(SL(n,\mathbb {Z})\)  on the \(n\) -torus \(\mathbb {T}^n\)  [1]}. It was used again by Katok, Lewis and Zimmer in various works [2]}, [3]}, [4]}, then by Katok and Spatzier in their work on local rigidity [5]}. As we will discuss below, these ideas were further developed by Goetze and Spatzier in [6]}.
